Pretty good measures in quantum information theory

TL;DR

This paper establishes a new inequality relating two quantum Renyi divergences, introduces a 'pretty good fidelity' concept, and provides conditions for optimality in quantum measurements and singlet fractions.

Contribution

It proves a reverse Araki-Lieb-Thirring inequality linking Petz and minimal quantum Renyi divergences, and introduces the 'pretty good fidelity' with implications for quantum measurement optimality.

Findings

New inequality between quantum Renyi divergences

Introduction of 'pretty good fidelity' concept

Conditions for optimality of quantum measurements

Abstract

Quantum generalizations of Renyi's entropies are a useful tool to describe a variety of operational tasks in quantum information processing. Two families of such generalizations turn out to be particularly useful: the Petz quantum Renyi divergence $\bar{D}_{\alpha}$ and the minimal quantum Renyi divergence $\tilde{D}_{\alpha}$. In this paper, we prove a reverse Araki-Lieb-Thirring inequality that implies a new relation between these two families of divergences, namely that $\alpha \bar{D}_{\alpha}(\rho \| \sigma) \leq \tilde{D}_{\alpha}(\rho \| \sigma)$ for $\alpha \in [0,1]$ and where $\rho$ and $\sigma$ are density operators. This bound suggests defining a "pretty good fidelity", whose relation to the usual fidelity implies the known relations between the optimal and pretty good measurement as well as the optimal and pretty good singlet fraction. We also find a new necessary and sufficient condition for optimality of the pretty good measurement and singlet fraction.